The limit of tanx/x as x approaches infinity does not exist, that is, limx→∞ tanx/x is undefined. Here we learn to find the limit of tanx/x when x tends to infinity.
The formula of the limit of tanx/x when x approaches ∞ is given by limx→∞ tanx/x = undefined. Below theorem will be useful to prove this formula.
Theorem: If limx→c f(x) exists, then for every sequence {xn} converging to c the limit limn→∞ f(xn) is always unique.
What is the Limit of tanx/x when x→∞
Answer: The value of the limit of tanx/x when x→∞ is undefined.
Proof:
Let f(x) = $\dfrac{\tan x}{x}$.
In order to show limx→∞ f(x) does not exist, we consider two sequences {xn} and {yn}, both converging to ∞, but f(xn) and f(yn) converge to different limits. Let us assume that
xn = nπ
yn = π/2 + nπ – 1/n2
Here, both xn, yn →∞, but we have that
limn→∞ f(xn) = limn→∞ $\frac{\tan n\pi}{n \pi}$ = 0.
On the other hand,
limn→∞ f(yn) = limn→∞ $\frac{\tan (\pi/2 + n\pi-1/n^2)}{\pi/2 + n\pi-1/n^2}$ = ∞.
As both sequences f(xn) and f(yn) converge to different limits, by the above theorem we conclude the following:
The limit of tanx/x when x→∞ does not exist.
ALSO READ:
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| limx→0 x/cosx = 0 | limx→0 x/sinx = 1 |
| limx→0 tanx/x = 1 | Sum rule of limits |
| Product rule of limits | Quotient rule of limits |
| ε – δ definition of limit |
FAQs
Q1: Find the limit of tanx/x as x tends to ∞.
Answer: The limit of tanx/x as x tends to ∞ is not defined.